Scrub the potato and carrot, then dice them into small half-inch cubes for even cooking.
Finely chop the yellow onion and mince the garlic clove.
Heat the olive oil in a large non-stick skillet over medium heat.
Add the diced potato, carrot, and onion to the skillet, seasoning with half of the sea salt and black pepper.
Sauté the vegetables for 10 to 12 minutes, stirring occasionally, until the potatoes are tender and golden brown.
Stir in the minced garlic and cook for an additional minute until fragrant.
Create four small wells in the vegetable hash and crack one egg into each opening.
Cover the skillet with a lid and cook for 3 to 5 minutes until the egg whites are set but the yolks remain runny.
While the eggs are cooking, finely dice the cucumber and stir it into the Greek yogurt along with the remaining salt and pepper.
Slide the hash onto plates and serve immediately with a generous dollop of the chilled cucumber yogurt.